Is ${210879}$ divisible by $9$ ?
Solution: A number is divisible by $9$ if the sum of its digits is divisible by $9$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {210879}= &&{2}\cdot100000+ \\&&{1}\cdot10000+ \\&&{0}\cdot1000+ \\&&{8}\cdot100+ \\&&{7}\cdot10+ \\&&{9}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {210879}= &&{2}(99999+1)+ \\&&{1}(9999+1)+ \\&&{0}(999+1)+ \\&&{8}(99+1)+ \\&&{7}(9+1)+ \\&&{9} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {210879}= &&\gray{2\cdot99999}+ \\&&\gray{1\cdot9999}+ \\&&\gray{0\cdot999}+ \\&&\gray{8\cdot99}+ \\&&\gray{7\cdot9}+ \\&& {2}+{1}+{0}+{8}+{7}+{9} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $9$ , so the first five terms must all be multiples of $9$ That means that to figure out whether the original number is divisible by $9 $ , all we need to do is add up the digits and see if the sum is divisible by $9$ . In other words, ${210879}$ is divisible by $9$ if ${ 2}+{1}+{0}+{8}+{7}+{9}$ is divisible by $9$ Add the digits of ${210879}$ $ {2}+{1}+{0}+{8}+{7}+{9} = {27} $ If ${27}$ is divisible by $9$ , then ${210879}$ must also be divisible by $9$ ${27}$ is divisible by $9$, therefore ${210879}$ must also be divisible by $9$.